Spectral Analysis of the Sum of Infinitesimal Perturbations in the Mathematical Models of Dynamic Systems

G.T. Arazov and T.H. Aliyeva

Abstract:

Continuous changes are happening in the dynamic systems observed in nature: time, configuration of objects and the mass of these objects. On the basis, the certain parameters of the system also change over time.

In this paper, we consider the sum of infinitesimal perturbations of the boundary values of observations used in mathematical models. For each time point, a specific set of numbers corresponds that can be determined from a comparative analysis of observation results and calculations by related models. Observations can be divided into the following parts:1) those that may be determined by mathematical modeling; 2) those that take hidden part in the observed processes. They usually are elusive. Over the time, they can cause a variety of resonance phenomenaor processes, such as chaos and catastrophes; 3) errors of the equipment used: measurement of time; measurement of the distance between the bodies of the system and their masses; 4) errors of performers of the work.

As an example, this paper presents the results of a comparative analysis of the coordinates α and δ, for resonant asteroids of Hecuba family. Their boundary values can be represented by the expressions:

−390s, 00 ≤ ε(ΔαTO) ≤ 615s02,

−1812’’00 ≤ ε(ΔδTO) ≤ 1029’’00; [10:22(1940−1962)] (I)

where, the number of observations subjected to comparative analysis and years covered by these observations are indicated in brackets. In the expressions (I), ε indicates compliance of this parameter to the parameter ε in the A.M. Lyapunov’s theorems on stability. It follows that as long as the sum of infinitesimal perturbations vary within the boundaries of (I), they are stable in the sense A.M. Lyapunov. In the case of violation of the borders (I), additional perturbation forces (or objects) are creeping into Solar system and they can cause a variety of resonance phenomena such as chaos or catastrophes.

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